Probability

John J. Kinney, PhD, is Professor Emeritus of Mathematics at Rose-Hulman Institute of Technology in Terre Haute, Indiana. A Member of the American Statistical Association and the Colorado Council of Teachers of Mathematics, Dr. Kinney is the author of numerous journal articles and three books, including A Probability and Statistics Companion, also published by Wiley.

Preface for the First Edition xi

Preface for the Second Edition xv

1. Sample Spaces and Probability 1

1.1. Discrete Sample Spaces 1

1.2. Events; Axioms of Probability 7

Axioms of Probability 8

1.3. Probability Theorems 10

1.4. Conditional Probability and Independence 14

Independence 23

1.5. Some Examples 28

1.6. Reliability of Systems 34

Series Systems 34

Parallel Systems 35

1.7. Counting Techniques 39

Chapter Review 54

Problems for Review 56

Supplementary Exercises for Chapter 1 56

2. Discrete Random Variables and Probability Distributions 61

2.1. Random Variables 61

2.2. Distribution Functions 68

2.3. Expected Values of Discrete Random Variables 72

Expected Value of a Discrete Random Variable 72

Variance of a Random Variable 75

Tchebycheff’s Inequality 78

2.4. Binomial Distribution 81

2.5. A Recursion 82

The Mean and Variance of the Binomial 84

2.6. Some Statistical Considerations 88

2.7. Hypothesis Testing: Binomial Random Variables 92

2.8. Distribution of A Sample Proportion 98

2.9. Geometric and Negative Binomial Distributions 102

A Recursion 108

2.10. The Hypergeometric Random Variable: Acceptance Sampling 111

Acceptance Sampling 111

The Hypergeometric Random Variable 114

Some Specific Hypergeometric Distributions 116

2.11. Acceptance Sampling (Continued) 119

Producer’s and Consumer’s Risks 121

Average Outgoing Quality 122

Double Sampling 124

2.12. The Hypergeometric Random Variable: Further Examples 128

2.13. The Poisson Random Variable 130

Mean and Variance of the Poisson 131

Some Comparisons 132

2.14. The Poisson Process 134

Chapter Review 139

Problems for Review 141

Supplementary Exercises for Chapter 2 142

3. Continuous Random Variables and Probability Distributions 146

3.1. Introduction 146

Mean and Variance 150

A Word on Words 153

3.2. Uniform Distribution 157

3.3. Exponential Distribution 159

Mean and Variance 160

Distribution Function 161

3.4. Reliability 162

Hazard Rate 163

3.5. Normal Distribution 166

3.6. Normal Approximation to the Binomial Distribution 175

3.7. Gamma and Chi-Squared Distributions 178

3.8. Weibull Distribution 184

Chapter Review 186

Problems For Review 189

Supplementary Exercises for Chapter 3 189

4. Functions of Random Variables; Generating Functions; Statistical Applications 194

4.1. Introduction 194

4.2. Some Examples of Functions of Random Variables 195

4.3. Probability Distributions of Functions of Random Variables 196

Expectation of a Function of X 199

4.4. Sums of Random Variables I 203

4.5. Generating Functions 207

4.6. Some Properties of Generating Functions 211

4.7. Probability Generating Functions for Some Specific Probability Distributions 213

Binomial Distribution 213

Poisson’s Trials 214

Geometric Distribution 215

Collecting Premiums in Cereal Boxes 216

4.8. Moment Generating Functions 218

4.9. Properties of Moment Generating Functions 223

4.10. Sums of Random Variables–II 224

4.11. The Central Limit Theorem 229

4.12. Weak Law of Large Numbers 233

4.13. Sampling Distribution of the Sample Variance 234

4.14. Hypothesis Tests and Confidence Intervals for a Single Mean 240

Confidence Intervals, 𝜎 Known 241

Student’s t Distribution 242

p Values 243

4.15. Hypothesis Tests on Two Samples 248

Tests on Two Means 248

Tests on Two Variances 251

4.16. Least Squares Linear Regression 258

4.17. Quality Control Chart for X 266

Chapter Review 271

Problems for Review 275

Supplementary Exercises for Chapter 4 275

5. Bivariate Probability Distributions 283

5.1. Introduction 283

5.2. Joint and Marginal Distributions 283

5.3. Conditional Distributions and Densities 293

5.4. Expected Values and the Correlation Coefficient 298

5.5. Conditional Expectations 303

5.6. Bivariate Normal Densities 308

Contour Plots 310

5.7. Functions of Random Variables 312

Chapter Review 316

Problems for Review 317

Supplementary Exercises for Chapter 5 317

6. Recursions and Markov Chains 322

6.1. Introduction 322

6.2. Some Recursions and their Solutions 322

Solution of the Recursion (6.3) 326

Mean and Variance 329

6.3. Random Walk and Ruin 334

Expected Duration of the Game 337

6.4. Waiting Times for Patterns in Bernoulli Trials 339

Generating Functions 341

Average Waiting Times 342

Means and Variances by Generating Functions 343

6.5. Markov Chains 344

Chapter Review 354

Problems for Review 355

Supplementary Exercises for Chapter 6 355

7. Some Challenging Problems 357

7.1. My Socks and √𝜋 357

7.2. Expected Value 359

7.3. Variance 361

7.4. Other “Socks” Problems 362

7.5. Coupon Collection and Related Problems 362

Three Prizes 363

Permutations 363

An Alternative Approach 363

Altering the Probabilities 364

A General Result 364

Expectations and Variances 366

Geometric Distribution 366

Variances 367

Waiting for Each of the Integers 367

Conditional Expectations 368

Other Expected Values 369

Waiting for All the Sums on Two Dice 370

7.6. Conclusion 372

7.7. Jackknifed Regression and the Bootstrap 372

Jackknifed Regression 372

7.8. Cook’s Distance 374

7.9. The Bootstrap 375

7.10. On Waldegrave’s Problem 378

Three Players 378

7.11. Probabilities of Winning 378

7.12. More than Three Players 379

r + 1 Players 381

Probabilities of Each Player 382

Expected Length of the Series 383

Fibonacci Series 383

7.13. Conclusion 384

7.14. On Huygen’s First Problem 384

7.15. Changing the Sums for the Players 384

Decimal Equivalents 386

Another Order 387

Bernoulli’s Sequence 387

Bibliography 388

Appendix A. Use of Mathematica in Probability and Statistics 390

Appendix B. Answers for Odd-Numbered Exercises 429

Appendix C. Standard Normal Distribution 453

Index 461